Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.777·2-s − 1.39·4-s + 0.222·5-s − 7-s − 2.63·8-s + 0.173·10-s + 6.52·13-s − 0.777·14-s + 0.738·16-s + 4.33·17-s + 2.91·19-s − 0.310·20-s + 3.89·23-s − 4.95·25-s + 5.07·26-s + 1.39·28-s + 3.77·29-s − 6.88·31-s + 5.85·32-s + 3.37·34-s − 0.222·35-s − 5.65·37-s + 2.26·38-s − 0.587·40-s − 1.33·41-s − 4.70·43-s + 3.03·46-s + ⋯
L(s)  = 1  + 0.549·2-s − 0.697·4-s + 0.0995·5-s − 0.377·7-s − 0.933·8-s + 0.0547·10-s + 1.81·13-s − 0.207·14-s + 0.184·16-s + 1.05·17-s + 0.668·19-s − 0.0694·20-s + 0.812·23-s − 0.990·25-s + 0.995·26-s + 0.263·28-s + 0.700·29-s − 1.23·31-s + 1.03·32-s + 0.577·34-s − 0.0376·35-s − 0.929·37-s + 0.367·38-s − 0.0928·40-s − 0.208·41-s − 0.717·43-s + 0.446·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.219507840$
$L(\frac12)$  $\approx$  $2.219507840$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 0.777T + 2T^{2} \)
5 \( 1 - 0.222T + 5T^{2} \)
13 \( 1 - 6.52T + 13T^{2} \)
17 \( 1 - 4.33T + 17T^{2} \)
19 \( 1 - 2.91T + 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 - 3.77T + 29T^{2} \)
31 \( 1 + 6.88T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + 1.33T + 41T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + 6.04T + 47T^{2} \)
53 \( 1 - 1.71T + 53T^{2} \)
59 \( 1 - 9.53T + 59T^{2} \)
61 \( 1 + 9.62T + 61T^{2} \)
67 \( 1 - 1.27T + 67T^{2} \)
71 \( 1 - 9.30T + 71T^{2} \)
73 \( 1 - 5.58T + 73T^{2} \)
79 \( 1 + 4.52T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 7.92T + 89T^{2} \)
97 \( 1 + 9.05T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.991256829994822170563748130676, −7.07199319089068958686626398889, −6.29030654927553765355059253344, −5.64959413633457286904005961081, −5.20349060346069109342414548661, −4.22668352078693290504276487353, −3.37524342248790485438630210689, −3.27313137801479796917678893332, −1.69661639267176711428418975807, −0.71916198273118171598871016749, 0.71916198273118171598871016749, 1.69661639267176711428418975807, 3.27313137801479796917678893332, 3.37524342248790485438630210689, 4.22668352078693290504276487353, 5.20349060346069109342414548661, 5.64959413633457286904005961081, 6.29030654927553765355059253344, 7.07199319089068958686626398889, 7.991256829994822170563748130676

Graph of the $Z$-function along the critical line